# One-way ANOVA

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One-way ANOVA

In statistics, one-way analysis of variance (abbreviated one-way ANOVA) is a technique used to compare means of two or more samples (using the F distribution). This technique can be used only for numerical data.

The ANOVA tests the null hypothesis that samples in two or more groups are drawn from the same population. To do this, two estimates are made of the population variance. These estimates rely on various assumptions (see below). The ANOVA produces an F statistic, the ratio of the variance calculated among the means to the variance within the samples. If the group means are drawn from the same population, the variance between the group means should be lower than the variance of the samples, following central limit theorem. A higher ratio therefore implies that the samples were drawn from different populations.

The degrees of freedom for the numerator is I-1, where I is the number of groups (means),e.g. I levels of urea fertiliser application in a crop. The degrees of freedom for the denominator is N - I, where N is the total of all the sample sizes.

Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test (Gosset, 1908). When there are only two means to compare, the t-test and the F-test are equivalent; the relation between ANOVA and t is given by F = t2.

## Assumptions

The results of a one-way ANOVA can be considered reliable as long as the following assumptions are met:

• Response variable must be normally distributed (or approximately normally distributed).
• Samples are independent.
• Variances of populations are equal.
• Responses for a given group are independent and identically distributed normal random variables (not a simple random sample (SRS)).

ANOVA is a relatively robust procedure with respect to violations of the normality assumption. If data are ordinal, a non-parametric alternative to this test should be used such as Kruskal-Wallis one-way analysis of variance.

• F test (Includes a one-way ANOVA example)

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